Politically it's unsustainable. If we see a half dozen more unicorns with household names people will tie completely private ownership to concerns about inequality: no way for the average person to buy a share and participate in growth.
Even if people widely consider the stock market to be an "equalizer", it doesn't seem obvious that adding more early-stage small-cap stocks to the market serves that purpose, because these companies are more likely to go bust, with a small fraction of them capturing most of the growth, and an index with a share in all those companies is not unlikely to underperform a large-cap index. (Incidentally, this is related to the reason that widespread stock ownership seems to me more likely to increase rather than decrease inequality - the prices of these things have a high variance; taxing and redistributing wealth and/or capital gains sounds like a surer path to equality, if that's what you want.)
Never underestimate the power of spin. Private just has to look worse than public on equality, and it does. I wouldn't be surprised to see an article about this in the next few weeks as journalists compete for eyeballs.
You are correct, people supporting capital markets do view/promote them as a method for companies to raise capital. If you look at the data though, this turns out to be misleading. The amount of money companies raise in initial or subsequent stock offerings through the public capital markets are a fraction of what they raise through other methods. Less than 7% of corporate investment capital is raised through stock offerings. Also, this is not just the case nowadays, it has been like this since the 1940s, and even before that.
It is not a bad idea by itself, and 7% is not an extraordinarily small number. In fact, it looks hight to me.
There are lots of problems to be found in the details, but the picture you paint isn't a dark one.